Twice-Universal Denoising
Erik Ordentlich, Krishnamurthy Viswanathan, M.J. Weinberger · IEEE Transactions on Information Theory · 2012
We propose a sequence of universal denoisers motivated by the goal of extending the notion of twice-universality from universal data compression theory to the sliding window denoising setting. Given a sequence lengthnand a denoiser, thekth-order regret of the latter is the maximum excess expected denoising loss relative to sliding window denoisers with window length 2k+1, where, for a given clean sequence, the expectation is over all channel realizations and the maximum is over all clean sequences of lengthn. We define the twice-universality penalty of a denoiser as its excesskth-order regret when compared to a bound on thekth-order regret of the denoising algorithm DUDE with parameterk, and we are interested in denoisers with a negligible penalty for allksimultaneously. We consider a class of denoisers that apply one of a number of constituent denoisers based on minimizing an estimated denoising loss and establish a formal relationship between the error in the estimated denoising loss and the twice-universality penalty of the resulting denoiser. Given a sequence of window parameterskn, increasing innsufficiently fast, we use this approach to construct and analyze a specific sequence of denoisers that achieves a much smaller twice-universality penalty forkknthan the sequence of DUDE denoisers with parameterkn.